Problem 82 - Brief Applied Calculus

Question

SOCIAL SCIENCE: Health Club Attendance A recent study analyzed how the number of visits a person makes to a health club varies with the monthly membership price. It found that the number of visits per year is given approximately by \(v(x)=-0.004 x^{2}+0.56 x+42\), where \(x\) is the monthly membership price. What monthly price maximizes the number of visits?

Step-by-Step Solution

Answer

The monthly price that maximizes visits is 70.

Step 1: Understand the function type

The function given is a quadratic function in the form of \(v(x)=-0.004x^2 + 0.56x + 42\). This type of function has a parabolic shape and opens downwards because the coefficient of \(x^2\) is negative. This implies there is a maximum point at the vertex of the parabola.

Step 2: Find the vertex formula

The vertex of a parabola given by \(ax^2 + bx + c\) occurs at \(x = \frac{-b}{2a}\). For the function \(v(x)\), \(a = -0.004\) and \(b = 0.56\). We will use this formula to find the \(x\) value that maximizes \(v(x)\).

Step 3: Apply the vertex formula

Plug the values of \(a\) and \(b\) into the vertex formula: \( x = \frac{-0.56}{2\times -0.004} = \frac{-0.56}{-0.008} = 70.\)

Step 4: Interpret the result

The result \(x=70\) refers to the monthly membership price. This means that the monthly membership price that maximizes the number of visits to the health club is \(70\) units of currency.

Key Concepts

Vertex Formula

The vertex formula is a crucial tool when working with quadratic functions. It helps pinpoint the specific point on a parabola called the vertex. Here's how it works:

• The formula for the vertex in a quadratic equation of the form \(ax^2 + bx + c\) is \(x = \frac{-b}{2a}\).
• This formula gives the x-coordinate of the vertex, which is the point of maximum or minimum value, depending on the parabola's direction.

For the problem where we have the equation \(v(x)=-0.004x^2 + 0.56x + 42\), the vertex formula finds the point where the maximum number of visits occurs. Because the coefficient \(-0.004\) of \(x^2\) is negative, the parabola opens downwards, indicating that this vertex is a maximum point. By inserting \(-0.004\) for \(a\) and \(0.56\) for \(b\) into the formula, we find that the vertex occurs at \(x = 70\). This step-by-step process highlights the equation's geometry and helps solve real-world problems effectively.

Maximization Problems

Maximization problems are a common application of quadratic functions. These problems often involve finding the best outcome or highest value for a situation described by a quadratic equation.

• Quadratic functions model scenarios where there is an optimal point resulting in a maximum (or minimum) value.
• In our exercise, the goal was to find the membership price that maximizes health club visits.

This process involves identifying a quadratic equation and applying it using calculus or algebraic methods like the vertex formula to find the optimal value. By plugging the coefficients \(a = -0.004\) and \(b = 0.56\) into the vertex formula, we derived that a membership price of 70 maximizes visits. Real-life scenarios, such as maximizing profits or attendance, can often be tackled efficiently with these mathematical techniques, providing valuable insights for decision-making.

Parabolic Functions

Understanding parabolic functions helps solve a wide range of mathematical problems, including our health club attendance problem. These functions have distinct characteristics that are important to grasp:

• A parabolic function is represented graphically as a curve known as a parabola.
• This parabola can open upwards or downwards, depending on the sign of the coefficient \(a\).
• If \(a\) is positive, the parabola opens upwards, and the vertex is a minimum point. If \(a\) is negative, as in our case, it opens downwards, and the vertex is a maximum point.

In the given equation \(v(x)=-0.004x^2 + 0.56x + 42\), the coefficient of \(x^2\) is \(-0.004\), leading the parabola to open downwards. Recognizing such forms is essential in interpreting data and trends effectively. Parabolas can reliably model both natural and engineered phenomena, allowing for accurate predictions and strategic planning.